The physical universe seems so complex and bewildering, it might appear a hopeless task for human beings to make any sense of it. Yet the enterprise called science is predicated on the audacious belief that, in spite of appearances, nature is both ordered and intelligible to us. Scientists assume that hidden beneath the surface complexity of the world lies a harmonious law-like simplicity. It is the job of science to uncover this hidden order, and thereby reveal the rules that govern the operation of the cosmos.

This ambitious project has been greatly facilitated by the philosophy of reductionism. In its commonly discussed form, reductionism is the hypothesis that complicated wholes are composed of simple parts, and that an understanding of the whole may be attained by knowledge of the individual component parts.

Reductionism is an ancient concept, dating from the Greek school of atomism. Democritus and Leucippus taught that the universe consists of nothing more than atoms moving in a void. The atoms were supposed to be truly primitive and indestructible entities, without internal parts. The immense richness and variety of material objects were accounted for by the different types of atoms they contain, and everything that happens in the universe is attributed simply to the rearrangement of atoms.

The objects that scientists today call atoms are not the basic level of matter that the Greeks had in mind. They are composite bodies with smaller pieces within them. Nevertheless, the belief that at some sufficiently minute scale of size there exists a small number of types of truly elementary entities out of which everything in nature is built remains as firm as ever. These basic objects may be subatomic particles like quarks, little loops of string, or something else entirely.

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More significant than the specific nature of the elementary entities is the prevalent belief that the quest for this bottom level of reality is indeed the proper way to uncover the "fundamental" workings of the universe. Pure reductionists reason that if physicists can discover all the properties of the basic building blocks of matter, then they will in principle have "explained" all of nature. Such schemes are often dubbed theories of everything, or final theories, and there is currently much speculation that scientists are on the verge of producing such a theory.

There is no doubt that, historically, reductionism as a methodology has played an important role in advancing science. In physics, a knowledge of the properties of atoms and subatomic particles has led to an understanding of the laws of chemistry, the thermal and electrical properties of materials and the source of power of the stars, among other things. In biology, the study of genes has explained a vast range of biological processes and medical conditions.

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The physicist Steven Weinberg has remarked that in science the arrows of explanation point downwards - down, that is, towards the simplest, most basic components of physical systems. Indeed, the very word "analysis", often used synonymously with rational scientific investigation, suggests that the best way to understand something is to break it apart into its constituents. Thus it is often said that, in principle, human behaviour can be reduced to biology, biology can be reduced to chemistry, chemistry to physics, and physics to the study of the elementary particles and fields of the subatomic realm.

The sweeping successes of reductionism as a scientific method have encouraged the belief that a reductionist picture of nature is the most satisfactory account of how the world actually is. In other words, you and I, and all else besides, are nothing but collections of subatomic particles interacting through blind and purposeless forces. Thus, according to this extreme "ontological" reductionism, once we know enough about elementary particles and forces we have in effect explained everything in the physical universe.

This bleak and austere philosophy - what Arthur Koestler dubbed "nothing-buttery" - has provoked an intense backlash from non-scientists, and some scientists and philosophers too. It seems to represent everything that is cold and soulless about science, stripping nature not only of its mystery, but many other important qualities too. If we are all nothing but meaningless mounds of atoms, what then of love, of human passion, of culture?

To appreciate the issue in bald terms, consider the example of living organisms. No atom of my body is alive, yet I am alive. Is the property of "being alive" a meaningful one requiring a scientific explanation, or is it merely a convenient label for some qualities of a complex system that ultimately can be entirely reduced to the properties of atoms? Reductionists say that the quality we call "life" is not a fundamental property at all, but a secondary quality that becomes apparent when a physical system becomes sufficiently complex. Anti-reductionists retort that this simply sweeps a real property of nature under the carpet, that you simply cannot capture the essence of what it means to be alive by focusing on the inanimate atoms of which an organism is composed.

In this collection of essays, leading scientists and philosophers from both sides of the debate attempt to inject some new life into this old controversy. Champion of the "hard reductionist" camp is Oxford chemist Peter Atkins, who uses the opportunity to launch a bitter attack on all forms of religion, which he regards as both antireductionist and deeply anti-scientific, a product of "ignorance, art and fear". By placing their trust in reductionism, claims Atkins, scientists are privileged to be at the summit of knowledge, able to sweep away the foggy sentimentality of other belief systems. Thus science reveals that the "wonderful richness of the world" is in fact nothing more than "growth from the dunghill of purposeless corruption", generated as the cosmos as a whole slides into a state of final chaos under the inexorable imperative of the second law of thermodynamics. But above all, because scientists are "the hewers of simplicity from complexity", their work leads us to appreciate the purposelessness of an existence that is understandable only in terms of a "fully-reduced simplicity".

This is powerful, pungent stuff. Atkins eloquently describes all that is repugnant to many non-scientists about the corrosive nature of rampant reductionist thought, and his challenge is taken up by the philosopher Mary Midgley in an essay aptly titled "Reductive megalomania". Midgley makes the important point that we do not have to choose between reductionism and holism, between atoms and humans. We can adopt a double-aspect view of reality wherein both parts and wholes are valid descriptions, being complementary rather than contradictory.

Midgley also raises the question of morality, which Atkins seems to have defined into non-existence. She points out that religion is not the prime casualty of reductivist thought: it is human individuality and associated moral responsibility. If only for this reason, Midgley implores that rampant reductionism be countered.

I agree with Midgley that Atkins's conflation of anti-reductionism with religion and superstition misses the point. True, religious thought is quite at odds with total reductionism. However, there are many non-religious scientists who are deeply sceptical that reductionism can yield a complete and satisfactory description of nature. Take, for example, the issue of causation. We can think of the world as a hierarchy of levels of complexity, with subatomic particles at the bottom and, say, human society near the top. Reductionism implies that just as the arrows of explanation point downwards, so the arrows of causation point upwards, so that, ultimately, everything that happens in the world is the result of atoms pushing and pulling on each other.

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There is an alternative, heretical viewpoint though, known as downwards causation, according to which some things that happen in a physical system can be explained only by taking into account the system as a whole. Downwards causation is suggested in highly complex self-organising systems. For example the human mind, representing a higher-level description, might exercise causative influence over the neurones of the brain, which belong to a lower-level description. This could explain how our wishes and decisions translate into bodily actions. Although such downwards causation has yet to be demonstrated, it is certainly physically possible, and need have nothing to do with religion. The world might simply function that way, with a sort of two-way traffic of causation and explanation, and neither "up" or "down" laying claim to being the more "fundamental".

It is, in fact, the nature of consciousness that Roger Penrose addresses in his essay. Developing on the theme of his famous book The Emperor's New Mind, he uses both mathematical and physical arguments to reject the reductionist notion that human minds and brains are nothing but the activities of elaborate computers. A key part of Penrose's thesis is that quantum physics, with its attendant non-locality, is important in the brain. Roughly speaking, nonlocality implies that spatially separated parts of the brain can by physically entangled, their behaviour can be made comprehensible only by considering the system as a whole. Generally speaking, quantum systems, which are normally associated with atoms and molecules but can be larger, cannot be decomposed into their basic parts without irreversibly destroying the original quantum state. Penrose suggests that reductionism fails in mental activity precisely because quantum mechanics bestows a sort of holistic, collective organisation within the network of neurons that constitute brains.

A complementary view of minds and brains is given by Gerald Edelman and Giulio Tomoni, and enthusiastically endorsed by neurologist Oliver Sacks. Edelman too rejects the idea of the brain as a computer, hard-wired by our genes, on which mental activity "runs" like conventional software. An infant brain apparently has a great deal of plasticity. During development, coherent neuronal activity emerges through a process of variation and selection, akin to Darwinism. Edelman claims that this theory can account for key features of higher-order brain functions, and serve as a model for artificial intelligence. As Sacks remarks, the attractive feature of Edelman's ideas is that they are grounded in biological reality, drawing on the developmental, anatomical and functional details of the nervous system, and responsive to the mental and physical experiences of the animal.

So much for the mind-brain problem, perhaps the thorniest of all the problems of reduction. But what about complete reduction - reducing the whole universe to a set of simple mathematical formulae that you might wear on a T-shirt? Two essays by cosmologist John Barrow and mathematician Gregory Chaitin examine the mathematical underpinning of physical reality. Can the world really be reduced to a set of formulae describing a simple, basic level of reality?

Barrow reviews recent attempts at producing such "theories of everything", stressing that a final theory should not only explain the properties of elementary particles and forces, but the existence of observers too. The latter, he points out, are intimately dependent upon the former. He conjectures that perhaps nature is actually an elaborate computational process rather than a system ruled by continuous mathematical laws. If so, it will be vastly more complicated than we thought, and the reductionist programme will have led us downwards from complexity, through relative subatomic simplicity, and finally back to enormous complexity again.

Taking up the theme of computational complexity, Chaitin reminds us that mathematics contains within itself the seeds of its own logical limitations. These limitations relate to the paradoxes of self-reference that Kurt Godel long ago showed can be used to prove a very deep result. Godel's theorem states that most consistent mathematical schemes, including ordinary arithmetic, are necessarily incomplete in the sense that certain propositions remain undecidable - they cannot be proved either true or false from within that system. Chaitin has shown that Godel incompleteness introduces an element of randomness even into mathematics. Consequently, the reductionist's dream of basing complex reality on simple mathematical formulae is undermined by an irreducible arbitrariness in mathematics itself.

Perhaps the final word should go to Freeman Dyson, a free-wheeling and philosophically minded physicist. Dyson enters an impassioned plea for scientists to renounce all "isms" anyway, and to use whatever tools and methods are necessary to make progress. Sometimes reductionism produces spectacular results, as with the discovery of quantum physics, but on other occasions a synthetic approach, often described as holism, works better, as when the study of the global properties of black holes helped elucidate the nature of Einstein's general theory of relativity.

My sympathies are entirely with Dyson. The proper goal of science is to explain the world, including phenomena such as life and consciousness. Explain means more than simply explain away. Occasionally reductionism gives a complete account of something at a higher level. More often, however, it remains a pious hope, a promise, a possibility "in principle". Attempts to divide physical phenomena into primary (reduced) and secondary (derived) are in my view simplistic. Reductionism as a research method should be pushed as far as it can go, but its dazzling successes must not blind us to the limited scope of this programme.

Paul Davies is professor of natural philosophy, University of Adelaide. His latest book is About Time: Einstein's Unfinished Revolution.